Here is the animation (2Mb) for my 9-piece symmetrical dissection,
which appears in Figure 2 of the article.
Notice that I emphasize the 3-fold rotational symmetry
about the axis that goes through the 1-cubes at opposite corners of the 7-cube.
Three times, I slide two rectangular blocks into relative position
and then rotate the group of pieces so far accumulated
120 degrees about this axis.
Once all six blocks are hovering just a bit outside of their eventual positions,
I slide them simultaneously into those positions.

The second identity in the family is

13 + 13 + 23 + 23 + 163 + 173 +183 = 193 + 203

Here is the animation (3Mb) for my 18-piece symmetrical dissection,
which appears in Figure 5 of the article.
Once again I emphasize the 3-fold rotational symmetry
in each of the resulting cubes, the 19- and 20-cubes.
Unfortunately, the animation system was for some reason, a bit uncooperative:
The 16-cube (the orange cube on the right) ends up obscuring
the 2-cube that slides behind it,
unlike what the 17-cube (the red cube on the left) does to the
1-cube that slides behind it.
Also, the slabs that are of thickness 2 end up obscuring
the 16-cube, unlike the 1-thick slabs and the 17-cube.
But hopefully you can still see (and imagine) what is going on.

There are dissections with fewer pieces than from the symmetric dissections. First:

13 + 13 + 53 + 63 = 73

Here is the animation (1.5Mb) for Robert Reid's 8-piece dissection,
which appears in Figure 7 of the article.
Notice that I have changed the dissection slightly from Figure 7,
to make it a bit easier to animate. The 2x6x6 block that contains
the stepped platform and the L-shaped piece is now on the bottom
rather than the top of the 6-cube.
Also, notice that the step in which I slide the L-shaped piece
away from the platform-shaped piece is not so easy to follow,
because of particular angle used in the perspective.
(Sorry, you'll just have to concentrate!)
A final comment: When you look at a perspective view of the resulting 7-cube,
you will never see all eight of the pieces simultaneously.
This is because when a cube is opaque,
you can see at most seven of its corners simultaneously,
and each of the eight pieces fills just one corner.

Let's also see what happens when we relinquish symmetry in a cube dissection for the second identity:

13 + 13 + 23 + 23 + 163 + 173 +183 = 193 + 203

Here is the animation (2.3Mb) for my 16-piece Reid-like dissection,
which appears in Figure 9 of the article.
It's a bit of a challenge to produce an effective animation,
because you (the viewer) must watch pieces that move back and forth across the screen,
which is a 2-dimensional projection of 3-space.
My solution was to have the 18-cube in the center, flanked by the 17-cube
on the left and the 16-cube on the right.
To build the 19- and 20-cubes simultaneously, I produce the 20-cube upside-down.
At any step in this process the scene on the screen more-or-less exhibits 180-degree rotational symmetry,
subject to the constraint that the 17-cube plus additional pieces give the 19-cube,
and the 16-cube plus additional pieces give the 20-cube.
In the final step, I rotate the resulting 20-cube to give its orientation as in Figure 9.

Producing the animations for this webpage impressed upon me
the fact that the symmetrical 9n-piece dissections
have pieces that are all rectangular blocks.
Indeed, for pieces made from cuts parallel to faces of the cube,
rectangular blocks are the only possible convex shapes.
On the other hand, the (arguably) minimal 8n-piece dissection
has 2n nonconvex pieces.
So we may well need to give up convexity
when we attempt a minimum number of pieces in the cube dissections for
our families of identities.
This seems reminiscent of the situation for the dissections of cubes for
33 + 43 + 53 = 63.
On page 201 of
Knotted Doughnuts and Other Mathematical Entertainments
(W. H. Freeman, New York, 1986),
Martin Gardner gave a convex 9-piece cube dissection discovered by Thomas H. O'Beirne.
Gardner stated that O'Beirne had shown that an 8-piece dissection
consisting of only rectangular solids was not possible.

September 25, 2009.

Text and animations are copyright 2009 by Greg Frederickson
and may not be copied, electronically or otherwise,
without his express written permission.